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17 Nov 2021, 07:00
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If x and y are positive integers , is \(x^y + (x^x)(y^y)\) odd?
(1) x + 2y is odd
(2) y + 2x is odd
(1) x + 2y is odd
(2) y + 2x is odd
17 Nov 2021, 07:37
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If x and y are positive integers , is x^y+(x^x)(y^y) odd?
Essentially asking two possibilities
1. If x is odd and y is even then x^y+(x^x)(y^y) becomes odd + odd x even = odd
or
2. If y is odd. In this case we have two cases
a. If x is even then x^y+(x^x)(y^y) becomes even + odd x even = even
b. If x is odd then x^y+(x^x)(y^y) becomes odd + odd x odd = even
Hence we can definitely prove that x^y+(x^x)(y^y) is not odd
irrespective of x being odd or even the answer is always even
(1) x + 2y is odd
Here x is odd but we don't know anything about y. Insufficient
(2) y + 2x is odd
Here y is odd.
As noted above, when y is odd, whether x is odd or even, the expression x^y+(x^x)(y^y) is always even. Hence this is sufficient to prove a Yes/No question
Option B
Essentially asking two possibilities
1. If x is odd and y is even then x^y+(x^x)(y^y) becomes odd + odd x even = odd
or
2. If y is odd. In this case we have two cases
a. If x is even then x^y+(x^x)(y^y) becomes even + odd x even = even
b. If x is odd then x^y+(x^x)(y^y) becomes odd + odd x odd = even
Hence we can definitely prove that x^y+(x^x)(y^y) is not odd
irrespective of x being odd or even the answer is always even
(1) x + 2y is odd
Here x is odd but we don't know anything about y. Insufficient
(2) y + 2x is odd
Here y is odd.
As noted above, when y is odd, whether x is odd or even, the expression x^y+(x^x)(y^y) is always even. Hence this is sufficient to prove a Yes/No question
Option B
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17 Nov 2021, 07:12
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Is \(x^y + (x^x)(y^y)\) odd?
For this to be odd, either of \(x^y\) & \((x^x)(y^y)\) needs to be odd and the other even.
But, since 'x' is present on both sides, 'x' can not be even, otherwise both the terms would be even. Thus, x has to be odd, and y has to be even. If we can prove this, it will answer our question: Is \(x^y + (x^x)(y^y)\) odd?
(1) x + 2y is odd.
For this to be odd, x has to be odd. We do not know the nature of y. y can be either even or odd. Insufficient.
(2) y + 2x is odd.
For this to be odd, y has to be odd. x can be even or odd. Insufficient.
(1) + (2),
x and y both are odd. Thus, \(x^y + (x^x)(y^y)\) will be even, because odd + odd = even.
Thus is sufficient to answer the question, Is \(x^y + (x^x)(y^y)\) odd? No. (C)
